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Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhvmulcan 27101 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) = (𝐴 · 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremhvmulcan2 27102 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremhvsubcan 27103 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremhvsubcan2 27104 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐶) = (𝐵 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremhvsub0 27105 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 0) = 𝐴)
 
Theoremhvsubadd 27106 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremhvaddsub4 27107 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 𝐶) = (𝐷 𝐵)))
 
19.1.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 27108 Inner product maps pairs from to . (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
·ih :( ℋ × ℋ)⟶ℂ
 
Theoremhicl 27109 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
 
Theoremhicli 27110 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ·ih 𝐵) ∈ ℂ
 
Axiomax-his1 27111 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 13549 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written 𝐴, 𝐵, but our operation notation co 6426 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4035. Physicists use 𝐵𝐴, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 27881. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
 
Axiomax-his2 27112 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
 
Axiomax-his3 27113 Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (𝐵 ·ih (𝐴 · 𝐶)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 27881 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
 
Axiomax-his4 27114 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))
 
19.2  Inner product and norms
 
19.2.1  Inner product
 
Theoremhis5 27115 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 · 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶)))
 
Theoremhis52 27116 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih ((∗‘𝐴) · 𝐶)) = (𝐴 · (𝐵 ·ih 𝐶)))
 
Theoremhis35 27117 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 · 𝐶) ·ih (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷)))
 
Theoremhis35i 27118 Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 · 𝐶) ·ih (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷))
 
Theoremhis7 27119 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 + 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶)))
 
Theoremhiassdi 27120 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))
 
Theoremhis2sub 27121 Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) − (𝐵 ·ih 𝐶)))
 
Theoremhis2sub2 27122 Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 𝐶)) = ((𝐴 ·ih 𝐵) − (𝐴 ·ih 𝐶)))
 
Theoremhire 27123 A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) ∈ ℝ ↔ (𝐴 ·ih 𝐵) = (𝐵 ·ih 𝐴)))
 
Theoremhiidrcl 27124 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
 
Theoremhi01 27125 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (0 ·ih 𝐴) = 0)
 
Theoremhi02 27126 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ·ih 0) = 0)
 
Theoremhiidge0 27127 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
 
Theoremhis6 27128 Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((𝐴 ·ih 𝐴) = 0 ↔ 𝐴 = 0))
 
Theoremhis1i 27129 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))
 
Theoremabshicom 27130 Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) = (abs‘(𝐵 ·ih 𝐴)))
 
Theoremhial0 27131* A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = 0 ↔ 𝐴 = 0))
 
Theoremhial02 27132* A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = 0 ↔ 𝐴 = 0))
 
Theoremhisubcomi 27133 Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 𝐵) ·ih (𝐶 𝐷)) = ((𝐵 𝐴) ·ih (𝐷 𝐶))
 
Theoremhi2eq 27134 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 𝐵)) = (𝐵 ·ih (𝐴 𝐵)) ↔ 𝐴 = 𝐵))
 
Theoremhial2eq 27135* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ 𝐴 = 𝐵))
 
Theoremhial2eq2 27136* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = (𝑥 ·ih 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremorthcom 27137 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 ↔ (𝐵 ·ih 𝐴) = 0))
 
Theoremnormlem0 27138 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ       ((𝐹 (𝑆 · 𝐺)) ·ih (𝐹 (𝑆 · 𝐺))) = (((𝐹 ·ih 𝐹) + (-(∗‘𝑆) · (𝐹 ·ih 𝐺))) + ((-𝑆 · (𝐺 ·ih 𝐹)) + ((𝑆 · (∗‘𝑆)) · (𝐺 ·ih 𝐺))))
 
Theoremnormlem1 27139 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝑅 ∈ ℝ    &   (abs‘𝑆) = 1       ((𝐹 ((𝑆 · 𝑅) · 𝐺)) ·ih (𝐹 ((𝑆 · 𝑅) · 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))))
 
Theoremnormlem2 27140 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))       𝐵 ∈ ℝ
 
Theoremnormlem3 27141 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   𝑅 ∈ ℝ       (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))))
 
Theoremnormlem4 27142 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   𝑅 ∈ ℝ    &   (abs‘𝑆) = 1       ((𝐹 ((𝑆 · 𝑅) · 𝐺)) ·ih (𝐹 ((𝑆 · 𝑅) · 𝐺))) = (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶)
 
Theoremnormlem5 27143 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   𝑅 ∈ ℝ    &   (abs‘𝑆) = 1       0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶)
 
Theoremnormlem6 27144 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   𝐴 = (𝐺 ·ih 𝐺)    &   𝐶 = (𝐹 ·ih 𝐹)    &   (abs‘𝑆) = 1       (abs‘𝐵) ≤ (2 · ((√‘𝐴) · (√‘𝐶)))
 
Theoremnormlem7 27145 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   (abs‘𝑆) = 1       (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))))
 
Theoremnormlem8 27146 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 + 𝐵) ·ih (𝐶 + 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶)))
 
Theoremnormlem9 27147 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 𝐵) ·ih (𝐶 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) − ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶)))
 
Theoremnormlem7tALT 27148 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝑆 ∈ ℂ ∧ (abs‘𝑆) = 1) → (((∗‘𝑆) · (𝐴 ·ih 𝐵)) + (𝑆 · (𝐵 ·ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
 
Theorembcseqi 27149 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 27209. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((𝐴 ·ih 𝐵) · (𝐵 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) · (𝐵 ·ih 𝐵)) ↔ ((𝐵 ·ih 𝐵) · 𝐴) = ((𝐴 ·ih 𝐵) · 𝐵))
 
Theoremnormlem9at 27150 Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))))
 
19.2.2  Norms
 
Theoremdfhnorm2 27151 Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
 
Theoremnormf 27152 The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
norm: ℋ⟶ℝ
 
Theoremnormval 27153 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
 
Theoremnormcl 27154 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
 
Theoremnormge0 27155 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 0 ≤ (norm𝐴))
 
Theoremnormgt0 27156 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
 
Theoremnorm0 27157 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(norm‘0) = 0
 
Theoremnorm-i 27158 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴) = 0 ↔ 𝐴 = 0))
 
Theoremnormne0 27159 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
 
Theoremnormcli 27160 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (norm𝐴) ∈ ℝ
 
Theoremnormsqi 27161 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
 
Theoremnorm-i-i 27162 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       ((norm𝐴) = 0 ↔ 𝐴 = 0)
 
Theoremnormsq 27163 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
 
Theoremnormsub0i 27164 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((norm‘(𝐴 𝐵)) = 0 ↔ 𝐴 = 𝐵)
 
Theoremnormsub0 27165 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((norm‘(𝐴 𝐵)) = 0 ↔ 𝐴 = 𝐵))
 
Theoremnorm-ii-i 27166 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (norm‘(𝐴 + 𝐵)) ≤ ((norm𝐴) + (norm𝐵))
 
Theoremnorm-ii 27167 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (norm‘(𝐴 + 𝐵)) ≤ ((norm𝐴) + (norm𝐵)))
 
Theoremnorm-iii-i 27168 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ       (norm‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (norm𝐵))
 
Theoremnorm-iii 27169 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (norm‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (norm𝐵)))
 
Theoremnormsubi 27170 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (norm‘(𝐴 𝐵)) = (norm‘(𝐵 𝐴))
 
Theoremnormpythi 27171 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝐴 ·ih 𝐵) = 0 → ((norm‘(𝐴 + 𝐵))↑2) = (((norm𝐴)↑2) + ((norm𝐵)↑2)))
 
Theoremnormsub 27172 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (norm‘(𝐴 𝐵)) = (norm‘(𝐵 𝐴)))
 
Theoremnormneg 27173 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (norm‘(-1 · 𝐴)) = (norm𝐴))
 
Theoremnormpyth 27174 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((norm‘(𝐴 + 𝐵))↑2) = (((norm𝐴)↑2) + ((norm𝐵)↑2))))
 
Theoremnormpyc 27175 Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → (norm𝐴) ≤ (norm‘(𝐴 + 𝐵))))
 
Theoremnorm3difi 27176 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (norm‘(𝐴 𝐵)) ≤ ((norm‘(𝐴 𝐶)) + (norm‘(𝐶 𝐵)))
 
Theoremnorm3adifii 27177 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (abs‘((norm‘(𝐴 𝐶)) − (norm‘(𝐵 𝐶)))) ≤ (norm‘(𝐴 𝐵))
 
Theoremnorm3lem 27178 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℝ       (((norm‘(𝐴 𝐶)) < (𝐷 / 2) ∧ (norm‘(𝐶 𝐵)) < (𝐷 / 2)) → (norm‘(𝐴 𝐵)) < 𝐷)
 
Theoremnorm3dif 27179 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (norm‘(𝐴 𝐵)) ≤ ((norm‘(𝐴 𝐶)) + (norm‘(𝐶 𝐵))))
 
Theoremnorm3dif2 27180 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (norm‘(𝐴 𝐵)) ≤ ((norm‘(𝐶 𝐴)) + (norm‘(𝐶 𝐵))))
 
Theoremnorm3lemt 27181 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℝ)) → (((norm‘(𝐴 𝐶)) < (𝐷 / 2) ∧ (norm‘(𝐶 𝐵)) < (𝐷 / 2)) → (norm‘(𝐴 𝐵)) < 𝐷))
 
Theoremnorm3adifi 27182 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
𝐶 ∈ ℋ       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘((norm‘(𝐴 𝐶)) − (norm‘(𝐵 𝐶)))) ≤ (norm‘(𝐴 𝐵)))
 
Theoremnormpari 27183 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
 
Theoremnormpar 27184 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))))
 
Theoremnormpar2i 27185 Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((norm‘(𝐴 𝐵))↑2) = (((2 · ((norm‘(𝐴 𝐶))↑2)) + (2 · ((norm‘(𝐵 𝐶))↑2))) − ((norm‘((𝐴 + 𝐵) − (2 · 𝐶)))↑2))
 
Theorempolid2i 27186 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       (𝐴 ·ih 𝐵) = (((((𝐴 + 𝐶) ·ih (𝐷 + 𝐵)) − ((𝐴 𝐶) ·ih (𝐷 𝐵))) + (i · (((𝐴 + (i · 𝐶)) ·ih (𝐷 + (i · 𝐵))) − ((𝐴 (i · 𝐶)) ·ih (𝐷 (i · 𝐵)))))) / 4)
 
Theorempolidi 27187 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 27113. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ·ih 𝐵) = (((((norm‘(𝐴 + 𝐵))↑2) − ((norm‘(𝐴 𝐵))↑2)) + (i · (((norm‘(𝐴 + (i · 𝐵)))↑2) − ((norm‘(𝐴 (i · 𝐵)))↑2)))) / 4)
 
Theorempolid 27188 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 27113. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (((((norm‘(𝐴 + 𝐵))↑2) − ((norm‘(𝐴 𝐵))↑2)) + (i · (((norm‘(𝐴 + (i · 𝐵)))↑2) − ((norm‘(𝐴 (i · 𝐵)))↑2)))) / 4))
 
19.2.3  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 27189 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
+ ∈ AbelOp
 
Theoremhilid 27190 The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
(GId‘ + ) = 0
 
Theoremhilvc 27191 Hilbert space is a complex vector space. Vector addition is +, and scalar product is ·. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⟨ + , · ⟩ ∈ CVecOLD
 
Theoremhilnormi 27192 Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
ℋ = (BaseSet‘𝑈)    &    ·ih = (·𝑖OLD𝑈)    &   𝑈 ∈ NrmCVec       norm = (normCV𝑈)
 
Theoremhilhhi 27193 Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
ℋ = (BaseSet‘𝑈)    &    + = ( +𝑣𝑈)    &    · = ( ·𝑠OLD𝑈)    &    ·ih = (·𝑖OLD𝑈)    &   𝑈 ∈ NrmCVec       𝑈 = ⟨⟨ + , · ⟩, norm
 
Theoremhhnv 27194 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ NrmCVec
 
Theoremhhva 27195 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        + = ( +𝑣𝑈)
 
Theoremhhba 27196 The base set of Hilbert space. This theorem provides an independent proof of df-hba 26998 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        ℋ = (BaseSet‘𝑈)
 
Theoremhh0v 27197 The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       0 = (0vec𝑈)
 
Theoremhhsm 27198 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        · = ( ·𝑠OLD𝑈)
 
Theoremhhvs 27199 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        = ( −𝑣𝑈)
 
Theoremhhnm 27200 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       norm = (normCV𝑈)
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